Free Vibration Analysis of Functionally Graded Spherical Torus Structure with Uniform Variable Thickness along Axial Direction

Table 5

Results of FG spherical shell with uniform variable thickness subject to classical and elastic edge conditions (θ₀ = π/4, θ_ζ = π/2, h₀ = 0.05 m, R = 1 m, a = 1/b = 0.5/c = 1/p = 1).

Type	n	m	Edge conditions
			F–C	F–SD	F–SS	C–C	C–SD	C–SS	E1–E1	E2–E2	E3–E3
FGMI	1	1	0.8061	0.8515	0.7695	1.4743	0.8666	1.4093	0.7110	0.7212	0.2475
		2	1.0844	1.0406	1.0226	1.7257	1.4854	1.5449	1.1898	1.7130	0.5718
		3	1.7518	1.5909	1.5809	2.6423	2.0498	2.5978	1.7672	1.8102	1.4872
		4	2.1285	1.9103	2.1195	3.0188	2.6236	2.7635	2.5254	2.7277	1.7468
		5	2.4997	1.9103	2.4987	4.0681	2.9832	3.9842	2.9350	3.0070	2.9186
		6	3.0641	2.1570	2.7648	4.6160	4.1743	4.3795	4.4855	4.2229	3.0165
	2	1	0.4738	0.1189	0.4319	1.3466	1.0130	1.2877	1.0765	1.0010	0.5706
		2	1.1650	1.0189	1.0935	1.7804	1.5538	1.5854	1.3907	1.7687	0.7426
		3	1.8301	1.0938	1.6483	2.9556	2.0093	2.7002	1.8000	2.4582	1.7623
		4	2.4691	1.6660	2.4686	3.3752	2.6982	3.3419	2.9086	2.9094	2.2710
		5	2.9391	2.7779	2.7936	4.0456	3.5972	3.9935	2.9754	3.1337	2.9568
		6	3.1283	2.8807	2.9793	4.6017	4.2218	4.3309	4.5264	4.4187	3.7186
	3	1	0.3786	0.2948	0.3498	1.2871	1.0927	1.2245	1.1397	1.1053	0.8530
		2	1.2615	1.1340	1.1833	1.8613	1.6617	1.6668	1.8575	1.8406	1.1818
		3	1.9470	1.6207	1.7557	3.0503	2.2682	2.7777	1.9651	2.9568	1.8673
		4	2.6336	1.7632	2.6335	4.0005	2.7748	3.9978	3.0411	3.1673	2.9612
		5	3.1661	2.8741	2.8910	4.2552	4.1804	4.1878	3.4069	3.7308	3.1233
		6	3.8750	3.3357	3.8718	4.6471	4.3587	4.3904	4.5893	4.5535	4.2193
FGMII	1	1	0.8036	0.8425	0.7625	1.4689	0.8635	1.4241	0.7085	0.7188	0.2468
		2	1.0797	0.9655	1.0147	1.7168	1.4789	1.5221	1.1857	1.7041	0.5693
		3	1.7431	0.9655	1.5624	2.6349	2.0434	2.5781	1.7581	1.8044	1.4827
		4	2.1233	1.0453	2.1184	3.0038	2.6117	2.7764	2.5192	2.7204	1.7377
		5	2.4916	1.6326	2.4759	4.0550	2.9734	3.9230	2.9195	2.9916	2.9041
		6	3.0495	2.1457	2.7919	4.5957	4.1517	4.4147	4.4618	4.2061	3.0081
	2	1	0.4722	0.1183	0.4224	1.3414	1.0103	1.2975	1.0719	0.9970	0.5690
		2	1.1600	1.0160	1.0894	1.7711	1.5469	1.5699	1.3870	1.7594	0.7390
		3	1.8209	1.0897	1.6321	2.9406	2.0032	2.6845	1.7906	2.4509	1.7532
		4	2.4612	1.6581	2.4575	3.3662	2.6846	3.3483	2.9013	2.8995	2.2647
		5	2.9311	2.7631	2.7934	4.0339	3.5870	3.9432	2.9598	3.1210	2.9416
		6	3.1140	2.8737	2.9857	4.5796	4.2000	4.3519	4.5027	4.3972	3.7091
	3	1	0.3767	0.2933	0.3434	1.2819	1.0908	1.2318	1.1348	1.1006	0.8490
		2	1.2561	1.1314	1.1845	1.8516	1.6538	1.6569	1.8477	1.8308	1.1780
		3	1.9371	1.6153	1.7440	3.0338	2.2623	2.7635	1.9599	2.9425	1.8581
		4	2.6255	1.7556	2.6188	3.9898	2.7606	3.9633	3.0247	3.1576	2.9462
		5	3.1501	2.8592	2.8917	4.2447	4.1618	4.1886	3.3985	3.7214	3.1145
		6	3.8656	3.3272	3.8656	4.6241	4.3439	4.3944	4.5654	4.5302	4.2081